Evidence on the Speed of Convergence to Market Efficiency by Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam April 11, 2004

Abstract Daily returns for stocks listed on the New York Exchange (NYSE) are not serially dependent. In contrast, order imbalances on the same stocks are highly persistent from day to day. These two empirical facts can be reconciled if sophisticated investors react to order imbalances within the trading day by engaging in countervailing trades sufficient to remove serial dependence over the daily horizon. How long does this actually take? The pattern of intra-day serial dependence, over intervals ranging from five minutes to one hour, reveals traces of efficiency-creating actions. For the actively-traded NYSE stocks in our sample, it takes longer than five minutes for astute investors to begin such activities. By thirty minutes, they are well along on their daily quest.

Contacts Chordia Roll Subrahmanyam Voice: 1-404-727-1620 1-310-825-6118 1-310-825-5355 Fax: 1-404-727-5238 1-310-206-8404 1-310-206-5455 E-mail: [email protected] [email protected] [email protected] Address: Goizueta Business School Anderson School Anderson School Emory University UCLA UCLA Atlanta, GA 30322 Los Angeles, CA 90095-1481 Los Angeles, CA 90095-1481

We are grateful to Michael Brennan, Jeff Busse, Eugene Fama, Laura Frieder, Will Goetzmann, Clifton Green, Andrew Karolyi, Pete Kyle, Francis Longstaff, Steve Ross, Ross Valkanov, Kumar Venkatraman, Ingrid Werner, and seminar participants at Bocconi University, Princeton University, Southern Methodist University, Texas A&M University, the New York Stock Exchange, the 2002 Western Finance Association Conference, and the 2002 University of Maryland Finance Conference for valuable comments and suggestions. We also appreciate financial support in the form of a grant from the Q-Group.

Evidence on the Speed of Convergence to Market Efficiency

2 Convergence to Efficiency, April 11, 2004 Evidence on the Speed of Convergence to Market Efficiency

Introduction

For most of its scientific life, the field of finance has debated the question of market efficiency.

Despite a long list of empirical anomalies and extensive indications of psychological quirks among investors, most financial economists and professionals still profess that asset prices are difficult to predict. Schwert (2001) reviews a number of well-documented anomalies and finds that some of them have disappeared, perhaps revealing ephemeral market inefficiencies. But he argues also that other anomalies appear to have been “discovered” even though they did not exist.

There is a growing literature about the irrationalities of individual investors. Odean (1999), for instance, finds that small investors have a perverse ability to forecast future returns; their stock purchases perform worse than their sales. Barber and Odean (2000) find that the more individuals trade, the worse their returns. Benartzi and Thaler (2001) document bizarre portfolio choices among individuals allocating pension assets to various classes.

Despite their reluctance to forecast prices, most scholars admit also that some individuals behave foolishly all the time and all individuals behave foolishly some of the time. When reconciling these conflicting views, we usually resort to flurry of hand waving and invoke the mantra of aggregation. Somehow, from within the blizzard of behavioral proclivities, the “market” becomes efficient, or, at least efficient enough that professors and money managers have a very

Convergence to Efficiency, April 11, 2004 3 difficult time beating passive investment strategies. But exactly how does this happen and how long does it take?

The concepts of market efficiency as defined by Fama (1970) in his seminal review, weak, semistrong, or strong form efficiency, represent a road map for statistical tests. They offer little insight about market processes that might deliver the hypothesized phenomena. Clearly, efficiency does not just congeal from spontaneous combustion. It depends, somehow, on individual actions.

This idea was formalized by Grossman (1976) and Grossman and Stiglitz (1980) who proved that the market price cannot fully incorporate all knowable information. Someone must be able to make (infra-marginal) returns from exploiting deviation of prices from fundamental values.

But whom, and how? Cornell and Roll (1981) borrowed a model from evolutionary biology to show that efficient markets must be inhabited by both passive investors, who take prices as correct forecasts of future value, and by active investors who expend resources in an effort to detect errors in prices. Market efficiency is the state in which neither the marginal active nor the marginal passive investor has an incentive to alter his or her respective approach. Infra-marginal active investors pay to become better informed and somehow move prices enough that passive investors can enjoy a free ride without sacrificing much return (indeed, any return at the margin).

Many investors still follow technical trading strategies that appear to generate little revenue and much cost; these strategies have long been the subject of much critique by finance professors.

Recently, Chordia, Roll, and Subrahmanyam (2002) document a seemingly related and intriguing

4 Convergence to Efficiency, April 11, 2004 phenomenon during a study of market-wide order imbalances on the New York Stock Exchange.

Market order imbalance, defined as the aggregated daily market purchase orders less sell orders for stocks in the S&P500 index, is highly predictable from day to day. A day with a high imbalance on the buy side will likely be followed by several additional days of aggregate buy side imbalance; and similarly for an imbalance on the sell side. This implies that investors continue buying or selling for quite a long time, either because they are herding or because they are splitting large orders across days, or both. More than fifty percent of tomorrow’s imbalance among S&P500 stocks can be forecast by past returns and past imbalances.

Yet the S&P500 index is virtually a random walk over a horizon of one day. During the 1996-

2002 sample period, it had a first order autocorrelation coefficient of -0.0015 (p-value=0.95) and insignificant autocorrelations at all longer daily lags. This suggests, of course, that some astute investors must be correctly forecasting continuing price pressure from order imbalances and conducting countervailing trades within the very first day, trades sufficient to remove all serial dependence in returns, which would otherwise be induced by the continuing procession of order imbalances.

There are at least two puzzles here: First, why do some naïve investors persist in their orders for days on end when it does them no good (because there is no inter-day return dependence)?

Second, how long within the day does pressure from order imbalances continue to move prices?

When thinking about this second and more important question, it seems rather obvious that some finite time period, albeit perhaps quite a short period, is required for sophisticated investors to counteract a sudden and unexpected preponderance of orders on the same side of the market.

Convergence to Efficiency, April 11, 2004 5

It simply cannot be true that returns are independent from trade to trade or even from minute to minute. It must take at least some time for astute investors to figure out what is happening to orders, to ascertain whether there is new pertinent information about values, and to expunge any serial dependence remaining after prices adjust to their new equilibrium levels. The horizon over which this activity takes place is the object of our study. We propose to investigate how long it takes the market to achieve weak-form efficiency; i.e., how long it takes to remove return dependence.

Other researchers have investigated questions similar to the one we address, but in very specific contexts. In early work, Patell and Wolfson (1984) show that dividend and earnings announcements “interrupt” the usual pattern of return serial dependence for at least fifteen minutes and that prices do not revert completely to their normal serial correlation pattern for up to ninety minutes. Although they make no explicit statement about how this happens, they clearly have in mind the activities of arbitrageurs who offset the impulsive reactions to company announcements of naïve investors.

Garbade and Lieber (1977) formulate a model of independent changes in equilibrium price coupled with random orders to buy or to sell at quoted ask and bid prices. They use data on two stocks for a single month and find that this model does not describe price moves for short time intervals (a few minutes) while it is consistent with price moves over longer horizons.1 In

1 Unlike us, Garbade and Lieber (1977) do not have access bid-ask quote mid-points and hence are unable to separate bid-ask bounce in transaction prices from true serial correlation.

6 Convergence to Efficiency, April 11, 2004 concluding, they recognize that “…investors who monitor the market continually during the day…” might be instrumental in bringing about the observed pattern.

Epps (1979) studies price adjustments for a group of firms in the same industry (automobiles).

He finds rapid but not instantaneous adjustments across firms to common news relevant for all industry firms. Correlations among the returns increase with the time interval, which suggests cross-firm variation in the speed of adjustment to new information. Epps’ overall conclusion is that “…the predictive value of a price change in one stock endures not much more than one hour…” but “…the average lag in the response of prices [to new information] is more then 10 minutes” (p. 298).

Related theoretical models were developed by Copeland (1976) and Hillmer and Yu (1979).

Copeland’s model predicts a positive correlation between trading volume and absolute price change and positive skewness in volume. However, it does not include a provision for the activities of arbitrageurs. Hillmer and Yu note that the incorporation of information into prices

“cannot be completed instantaneously” because “…in practice an investor will not react…unless he is convinced that it is economically advantageous.” (p. 321). They develop various alternative statistical models involving price, volume, and volatility, all inspired by the idea that investor/arbitrageurs would be watching the market closely and reacting occasionally. Their tests, however, involve only a handful of anecdotal events.

Much later, Chakrabarti and Roll (1999) formulate a model populated by Bayesian traders/arbitrageurs who attempt, through observing the trading of others, to deduce the quality

Convergence to Efficiency, April 11, 2004 7 of their information. Simulations of the model show that the market usually converges more rapidly to an equilibrium price and that it is a better predictor of true value when arbitrageurs react to one another as opposed to trading solely on their own information.

Section I below presents a theoretical framework to motivate our analysis. Section II describes the data. Section III presents our analysis of how quickly prices of highly liquid stocks become efficient. Section IV concludes and suggests further investigations.

I. A Theoretical Framework

Our goal is to analyze how quickly the market accommodates autocorrelated order flows. In order to motivate our empirical analysis, we present a multiperiod model that captures the economic rationale of our empirical tests. In the model, a risk averse market making sector dynamically accommodates liquidity demands (that are allowed to be autocorrelated).

Consider a security that is traded at each of two dates, 1 and 2, and pays off a random amount

θ+ε at date 3. The market consists of risk-averse market-making agents who absorb the orders emanating from liquidity traders. The market makers have exponential utility with risk aversion coefficient R. There are two types of liquidity orders. The first type arrives solely at date 2 and is denoted z2. The total quantity of the second type of liquidity orders is z1, but a fraction k of this order arrives at date 1 and the balance arrives at date 2. The total mass of the market makers is normalized to a number N and a mass M arrives at date 1 with the balance (N-M) arriving at date

8 Convergence to Efficiency, April 11, 2004 2. The variable θ is learned at date 2 and no information about the asset’s payoff is available at

date 1. In the analysis below, vX denotes var(X), except that z1 and z2 have a common variance

vz. All random variables are normally distributed with mean zero and are mutually independent.

Standard mean-variance analysis indicates that the date 2 demands of the market makers can be written as

θ − P2 x2 = Rvε

From the market clearing condition Nx2 + z1 + z2 = 0, we find that

Rv P = θ + ε (z + z ) 2 N 1 2

Consider now the date 1 equilibrium. The appendix shows that the date 1 demands of the market makers are given by

E(P2 | z1 ) − P1 S1 − S2 x1 = + E(x2 | z1 ) (1) RS1 S1

where S1 and S2 are the first and second elements, respectively, in the first row of the

matrix given by

−1 −1  2 2 2  vθ + (R vε / N )vz vθ   1/ vε −1/ vε    +    v v −1/ v 1/ v   θ θ   ε ε 

-1 with the first matrix simply representing cov(P2, θ|z1) . The expression in Eq. (1) represents a

Convergence to Efficiency, April 11, 2004 9 component to exploit the expected price appreciation across dates 1 and 2 as well as a second term to account for intertemporal hedging.

The market clearing condition at date 1 is given by Mx1 + kz1 = 0. Substituting for x1 and solving for P1, we find that

2 2 R[kN{R vε vz (vε + vθ ) + N vθ }+ MNvε ] P1 = 2 2 M (R vε vz + N )

Let Q1 = z1 represent the date 1 imbalance. We then have that

2 2 2 2 2 Rvz [MR v ε vz − kN{R v ε vz (vε + vθ ) + N vθ }] cov(P2 − P1 ,Q1 ) = 2 2 MN(R vε vz + N )

Thus, there is a positive predictive relation between price changes and imbalances so long as the mass of market makers at date 1 is sufficiently large. The notion is that autocorrelation in imbalances, coupled with inventory effects, cause the predictive relation. Intuitively, an imbalance in one direction predicts price changes because it is accompanied by further imbalances, and consequently further price pressure in the same direction. The predictive relation disappears as the risk aversion of the market makers (R) approaches zero, so that inventory effects become negligible. The predictive relation also disappears as the market making capacity of the market becomes large (i.e., as M and N become infinitely large). Hence tests of a predictive relation between imbalances and future returns are direct tests of inventory effects.

Our goal is to determine the horizon at which the market making capacity becomes large enough to cause the predictive relation to disappear. Thus, our notion of the speed of convergence to efficiency, in essence, captures the efficacy with which the market making sector accommodates

10 Convergence to Efficiency, April 11, 2004 imbalances from outside investors.

II. The Data

Since we already know that serial dependence in returns is close to zero for active stocks over a daily horizon, our investigation of the efficiency-creating process must focus on intra-day trading. We would like to measure the timing of efficiency creation as precisely as possible, so it seems sensible to examine frequently-traded stocks for which very short term serial dependence can actually be observed. This suggests that very small stocks should be excluded owing to the difficulty inherent in measuring serial dependence even when trading is infrequent.

Because transactions data are so voluminous (e.g., IBM alone has several million transactions a year), this study uses a limited sample of stocks and time. Our calculations here cover 150 large stocks listed on the New York Exchange for three recent years, 1996, 1999 and 2002. These years were chosen because (a) transactions data are available from the TAQ (Trade and

Automated Quotations) database recorded by the Exchange, and (b) they bracket significant changes in the minimum tick size, which was reduced from $1/8 to $1/16 during 1997 and was reduced further to one cent in by January 2001. We hoped to discern changes in the price formation process during years preceding and following these events. Future investigations should extend the investigation to smaller firms, and other years, exchanges, and countries.

Convergence to Efficiency, April 11, 2004 11 The 150 sample firms are listed in Table 1. The first fifty are the largest listed firms at the beginning of each sample year. Their market capitalizations (across the three sample years) range from $398 billion to $18.4 billion to 1996. The mid-cap group (i.e., mid-cap within our sample) consists of the next largest 50 stocks by market capitalization at the beginning of each year. Their sizes range from $41.0 billion to $9.72 billion. Finally, the smaller size group is comprised of 50 stocks with market capitalizations ranging from $18.8 billion to $6.69 billion.

Market caps within our sample vary over a range of about 60 to 1 but even the smallest firms are relatively large and should be actively trading.

Each transaction for each of the 150 stocks during the three years is recovered from the TAQ database, which provides not only trade prices, but also bid and ask quotes associated with each transaction. This allows us to use the Lee/Ready (1991) trade assignment algorithm to estimate whether a particular trade was buyer- or seller-initiated.2 Order imbalance for each stock over any time interval can then be calculated variously as the number of buyer- less the number of seller-initiated trades (OIB#), the number of buyer-initiated shares purchased less the number of seller-initiated shares sold (OIBSh), or the dollars paid by buyer-initiators less the dollars received by seller-initiators (OIB$).

The first of these order imbalance measures disregards the size of the trade, counting small orders equally with large orders. The second and third measures weight large orders more heavily. The distinction is important here because we hope to shed light on how arbitrageurs

2 The Lee/Ready algorithm classifies a trade is as buyer- (seller-) initiated if it is closer to the ask (bid) of the prevailing quote. The quote must be at least five seconds old. If the trade is exactly at the mid-point of the quote, a “tick test” is used whereby the trade is classified as buyer- (seller-) initiated if the last price change prior to the trade is positive (negative). Note that a limit order is most often the passive side of the trade; i.e., the non-initiator.

12 Convergence to Efficiency, April 11, 2004 make the market more efficient over very short horizons and presume that arbitrageurs tend to undertake larger trades as compared to naïve investors in order to quickly exploit deviations of prices from fundamentals.

III. The Evidence.

III.A. Evidence of efficiency at a daily horizon.

Using CRSP returns data,3 we first set out to ascertain whether our sample of stocks conformed to semistrong-form efficiency over a daily horizon; i.e., whether future returns could be predicted by either past returns or past order imbalances. Table 2 documents the daily return serial correlations and shows that the average first-order daily autocorrelation coefficient for the largest

50 stocks during 1996 was 0.005; the t-statistic, 0.48, was calculated from the cross-section of sample autocorrelation coefficients assuming independence.4

This positive (though insignificant) coefficient is somewhat surprising because negative first- order autocorrelation in trade-to-trade returns is known to be induced by the bid-ask bounce.

During 2002, the large stocks did exhibit such negative autocorrelation as did the mid-cap stocks in 2002 and the smaller 50 stocks in all years. No serial correlation coefficient is positively significant for any size group in any year

3 From the Center for Research in Securities Prices (CRSP) of the University of Chicago. 4 It seems likely that the assumption of cross-sectional independence actually results in an overstatement of statistical significance because returns, and hence sample correlation coefficients, are mostly positively correlated. This implies that the estimated standard error of the sample mean is too small since it omits the mostly positive covariance terms that would be in the true standard error.

Convergence to Efficiency, April 11, 2004 13 To avoid contamination of return serial correlations by bid-ask bounce, we compute returns from quote midpoints as well as from transaction prices. So, for each transaction during every day, the quotes existing at least five seconds before the trade were used to compute a bid-ask midpoint. Returns were then computed from these midpoints. For example, the daily midpoint returns in Table 2 are computed from the bid and ask quotes just prior to the last transaction of the day. Again, no positive coefficient is significant. The largest and mid-cap groups have significantly negative but small coefficients, -.028 and -.038, in 2002.

Table 2 also reports simple correlations between returns and the three measures of order imbalance, both contemporaneous correlations and correlations with OIB lagged by one day. As could be expected, there is a very strong positive contemporaneous correlation between either measure of return (trade or midpoint) and any of the OIB measures. Not surprising also, the share and dollar measures, OIBSh and OIB$, are considerably more highly correlated with contemporaneous returns, particularly for larger firms.

The correlations between daily returns and lagged (by one day) order imbalances are insignificant for the share and dollar imbalances of the largest and mid-cap firms. There is, however, significant negative correlation among the smallest 50 firms in 1999 though not in the other years.

Lagged OIB# (in shares) is significantly correlated with returns in all years for the largest 50 firms, in 1996 and 1999 for the mid-cap 50 and in 1996 for the smallest 50. The magnitude of the correlation is 0.07 or less, so the economic value of the implied prediction would be

14 Convergence to Efficiency, April 11, 2004 relatively small. Moreover, in each case the magnitude has declined over time. This is consistent with a small improvement in market efficiency perhaps brought about by the minimum tick size reduction.

Notice that the order imbalance measures themselves are strongly and positively autocorrelated from day to day, a feature particularly striking for OIB# (which weights all trades equally regardless of size). For the large stock group, the autocorrelation coefficient for OIB# exceeds

0.4 in both 1996 and 1999. It is 0.317 in 2002. The corresponding coefficients are somewhat smaller for smaller firms but they are positive and significant. In an earlier paper, Chordia, Roll, and Subrahmanyam (2001) show that even aggregate market order imbalances persist for several days.

III.B. Evidence about efficiency over short horizons with the trading day.

We computed short-horizon returns from prices closest to the end of various time intervals within the trading day. For example, ten-minute returns are computed for each stock by finding the transaction closest to 9:40 a.m., 9:50 a.m., etc.5. Since some calculations involve lagged values, the first interval of each trading day is discarded because it would have been correlated with a lagged interval from the previous trading day.6 Throughout this sub-section, all the reported regression coefficients were first computed within the trading day for each stock, then averaged across all trading days and stocks.

5New York Stock Exchange trading hours are 9:30 a.m. to 4:00 p.m. except for rare exceptions (e.g., 9/11). 6 Intervals of sixty minutes were set backward from the end of the trading day. For example, each day has five one- hour intervals (11-12, 12-1,…,3-4) included in the calculations; the interval from 10 to 11 a.m. provides lagged observations only and data from 9:30 to 10 a.m. are not used at all.

Convergence to Efficiency, April 11, 2004 15

There is admittedly some imperfection involved in computing very short-term returns because trades do not necessarily occur at the exact ending time of each interval. If the closest price to the end of an interval was more than 150 seconds away, either before or after, the return for that interval was not used in our calculations. Within the large stock sample, the average time between transactions was 19 seconds (averaged across the three years). Over intervals longer than five minutes, this problem obviously becomes progressively less material.

Order imbalances were computed over all trades within each time interval. For example, contemporaneous OIB# during the ten-minutes ending at 9:50 a.m. consists of the number of buyer-initiated trades less the number of seller-initiated trades between 9:40:01 a.m. and 9:50:00 a.m. The lagged ten-minute OIB# is the corresponding accumulation between 9:30:01 a.m. and

9:40:00 a.m.

We use two measures of significance. The first is the cross-sectional average of the regression t- statistic, and the second is computed from the cross-section of coefficients after accounting for cross-correlation in the regression residuals. The specific formula for the correction that we employ, also mentioned in footnote 8 of Chordia, Roll, and Subrahmanyam (CRS) (2000), assumes that the residual cross-correlation and the residual variance are homogeneous across stocks. Under these assumptions, an estimate of the amount by which the standard error is inflated is given by [1+(N-1)ρ]1/2, where N is the number of regressions (the CRS formula contains an erroneous numeral 2). In the formula, ρ is the common cross-correlation in the

16 Convergence to Efficiency, April 11, 2004 residuals, which is proxied by the average residual cross-correlation across the 50 adjacent regressions for stocks, sorted by alphabetical order.

Our first results are in Table 3, which reports serial regressions for returns and univariate regressions of returns on lagged order imbalances. Turning first to returns, there is little evidence of unconditional serial dependence. We base this conclusion on the second of the two reported t-statistics. Across all years and firm sizes, only one t-statistic out of 45 exceeds 2.0 in absolute value, slightly less than one would expect just by chance; 34 of the 45 are less than 1.0 in absolute value. This suggest that these stocks conform well to weak-form efficiency; i.e., using only the past history of returns, there is little, if any, predictability of future returns even over intervals as short as five minutes.

The story does not stop there, however, because Table 3 also shows that lagged order imbalances are often statistically significant predictors of future returns over short intervals. For the largest

50 stocks, there is significance at both five and ten minute intervals during 1996, though this declines to marginal significance in 1999 and to insignificance in 2002 (based on the second of the two t-statistics, which we believe is more reliable).

For mid-cap stocks, OIB displays significant predictive ability over five minutes in all years and over ten minutes during 1996 and 1999. Smaller stocks have a similar pattern and are even significant in some cases over 15-minute intervals in the two earlier years.

Convergence to Efficiency, April 11, 2004 17 The obvious pattern in all regressions where OIB predicts returns is the declining predictive ability as the return interval lengthens. The coefficients and t-statistics are much larger for the shorter intervals. This suggests that the market is not strong-form efficient over very short periods, perhaps as long as fifteen minutes for smaller companies. Strong-form efficiency is the appropriate criterion here because agents off the exchange cannot observe order imbalances easily; only the NYSE specialist and perhaps astute floor traders observe an imbalance immediately. However, traders seem able to deduce and accommodate the impact of an imbalance quickly and their accommodative capacities appear to have increased over time. In

2002, there is little predictive ability of OIB for large stocks even at five minutes and none for midcap and small stocks at ten minutes. In no year and for no size group does it take more than

30 minutes to eliminate the predictive content of OIB.

This is all the more impressive when one considers the serial regressions of OIB itself in Table

4. We saw earlier that OIB is strongly autocorrelated over a daily horizon (Table 2) and Table 4 confirms that a similar pattern is present even over intervals as short as five minutes. Indeed, the serial dependence in OIB actually increases as the interval lengthens from five minutes to one hour. Notice that the coefficients generally grow larger with interval length. The first t-statistic declines with interval length but this can be partly attributed to the smaller number of non- overlapping longer intervals. The second t-statistic, which is insensitive to the number of observations because it depends only on the cross-section of estimated coefficients, also usually declines with longer interval length, though not as much. The explanatory power (R-square) goes up modestly with interval length.

18 Convergence to Efficiency, April 11, 2004 In the absence of countervailing trading activity by arbitrageurs and the specialist, this persistence in order imbalances would have induced strong return predictive ability for OIB over longer intervals, which, as we have just seen, does not obtain. Consequently, the results are consistent with the notion that agents are acting not only to countervail imbalances concurrently but also to predict and forestall the influence of future imbalances.

Overall, we interpret these results to reveal the actions of three distinct groups. Order imbalances in the first instance arise from traders who demand immediacy for liquidity or informational needs. Order imbalances are positively autocorrelated, which suggests that traders are herding (Hirshleifer, Subrahmanyam, and Titman, 1994), or spreading their orders out over time (Kyle, 1985), or both. Second, NYSE specialists react to initial order imbalances by altering quotes away from fundamental value in an effort to control inventory. Finally, outside arbitrageurs (by way of market or limit orders) intervene to add market-making capacity by conducting countervailing trades in the direction opposite to the initial order imbalances. This arbitrage activity takes at least a few minutes.

Evidence supportive of outsider intervention is provided in Table 5, which correlates initial orders with future orders on the other side of the market. In other words, buyer-initiated orders are related to seller-initiated orders in the next time interval, and vice versa. Notice that every single coefficient is positive and strongly significant and that the coefficients increase both in size and in significance with interval length. The t-statistics are generally much larger in this table than they were for order imbalances in Table 4.

Convergence to Efficiency, April 11, 2004 19 To understand the combination of these results, notice that the serial covariance in order imbalances can be written and decomposed as follows:

Cov(OIBt,OIBt-1) = Cov(Bt-St,Bt-1-St-1)

= Cov(Bt,Bt-1) + Cov(St,St-1) - Cov(Bt,St-1) - Cov(St,Bt-1) where B and S denote buyer- and seller-initiated orders, respectively. Table 5 shows that the last two covariances are uniformly positive and growing with interval length. However, their growth with interval length is not sufficient to overcome the even stronger and growing persistence in

(naïve) orders of the same sign, which are captured in the first two covariances. Comparing

Tables 4 and 5, the significance of the serial dependence in OIB usually declines with interval length (Table 4) while the significance of countervailing trading increases substantially (Table

5).

Yet OIB is still positively dependent out through sixty minutes and even over an entire day.

Countervailing arbitrage never offsets it completely. Hence, to render the market informationally efficient, arbitrage trading must be augmented by assistance from two other phenomena, limit orders and specialist actions.

To trace these influences, Table 6 presents a series of multiple regressions with both lagged order imbalances and lagged returns as predictors.7 In all regressions, the future midpoint return is the dependent variable. Regressions were carried out for individual stocks and the table reports the average coefficients. Again, two t-statistics are provided. The first is simply the average individual coefficient’s t-statistic. The second is calculated from the cross-sectional array of

20 Convergence to Efficiency, April 11, 2004 estimated individual coefficients, correcting for cross-equation dependence. For a given intra- day return interval, all returns over the entire year are included in the regression except for the first interval return on each trading day. (It appears only as a lagged value.) For each return interval, Table 6 reports two regressions which differ by the measure of order imbalance used as a regressor, either OIB# for the number of trades OIB$ for the dollar amount traded. Both regressions include the lagged return.

Focusing first on 1996, lagged returns have significant negative coefficients in all regressions for five-, ten- and fifteen-minute intervals. Meanwhile, lagged OIB is significantly positive. For smaller stocks, lagged OIB remains significant even at thirty minutes but is only marginally significant for larger and midcap stocks. At sixty minutes, nothing is significant. When they are significant, OIB#t-1 is usually slightly more significant than OIB$t-1.

In 1999, there is a marked reduction in significance for all stock sizes. Lagged returns are not significant over any interval and lagged OIB is significant only out through fifteen minutes at most, (and that for small stocks and OIB#t-1). At five-minute intervals, OIB is still a significant positive predictor of future returns for all three size groups.

For 2002, order imbalances are significant at five minutes but at ten minutes only for large stocks and for mid-cap stocks using OIB#. Lagged returns are significant at five- and ten-minute intervals and even fifteen-minute intervals for larger stocks. Again, at longer intervals nothing is significant.

7 While there is clear multicollinearity induced by the inclusion of both contemporaneous and lagged imbalance, this should attenuate standard errors and reduce significance. Thus, multicollinearity does not detract from the

Convergence to Efficiency, April 11, 2004 21

Note that the Table 6 regressions are not evidence against weak-form market efficiency for very short intervals such as five minutes because outsiders cannot easily observe order imbalances. A trading rule based on the short-period results in Table 6 is therefore feasible only for the specialist and perhaps sophisticated floor traders.

It is perhaps worth digressing at this point to briefly examine the economic significance of our results. Considering the results for the five-minute interval during the year 2002, the coefficients

(scaled by 105) are respectively 1.07, 1.91, and 1.97. The cross-sectional averages of the standard deviations of the imbalance in number of transactions for the three samples (not reported in the tables) are 14.88, 11.73, and 8.83, respectively. Assuming an agent with knowledge of the imbalance trades once a day, the annualized return, assuming 250 trading days during an year, for three cases are 3.98%, 5.60%, and 4.44%. These numbers appear to be economically significant despite the low R2’s in the regressions, though the magnitude of the returns is not overwhelmingly large after accounting for transaction costs, especially for individual investors.

Overall, the results suggest that outsider arbitrage activity (as documented in Table 5), along with specialists’ actions and limit orders, increase market-making capacity and remove the influence of order imbalances within a half-hour and usually within a much shorter time. This rapid convergence to efficiency is all the more impressive when one considers that order imbalances persist strongly over much longer intervals, even over days. Evidently, astute agents

significant coefficients on which we focus.

22 Convergence to Efficiency, April 11, 2004 have little difficulty in forecasting the persistence in order imbalances and conducting trades sufficient to eliminate its impact on prices after just a few minutes.

The only remaining puzzle is why lagged returns are significantly negatively related to future returns in the multiple regressions of Table 6 when they are virtually unrelated to future returns in the univariate regressions of Table 4. At least two explanations seem possible and they are not mutually exclusive. First, prices might overreact to the initial order imbalance in t-1 and then rebound from the overreaction in t. Consequently, the return in t-1 is negatively related to the return in t (conditional on knowing OIBt-1). It would appear, however, that this behavioral explanation implies that the specialist is not trading optimally on his privileged information about OIB.

A second explanation is: the specialist responds to an initial order imbalance by intentionally adjusting the bid and ask prices by more than he knows is appropriate. This might help maintain control of inventory risk. By moving prices away from equilibrium, the specialist attracts outside arbitrageurs who enter and help “lean against the wind” of the highly persistent order imbalances. Notice that the negative coefficients of lagged returns, though significant at short intervals, are rather small in absolute magnitude. This suggests that the specialist is not leaving all that much money on the table. It might be well worth giving up a modest profit to attract the attention and aid of astute outsiders.

IV. Conclusions

Convergence to Efficiency, April 11, 2004 23 The long and continuing debate about financial market efficiency has been relatively silent about the behavior of actual traders. Somehow, perhaps unwittingly, they act collectively to push markets toward efficiency. Except in an idealized theoretical world, this cannot happen instantaneously. There must be some time interval, albeit very short, over which the actions of efficiency-creating traders remain incomplete. A central goal of this paper is to present evidence about this important issue, the speed of convergence to market efficiency.

We first examine weak-form efficiency (Fama, 1970), which is concerned only with serial dependence in returns. Of course, even weak-form efficiency might not be attained immediately.

But using intra-day returns for 150 NYSE stocks during calendar years 1996, 1999, and 2002, we find that weak-form efficiency does appear to prevail over intervals from five minutes to one day. There is evidence, however, that the market is not strong-form efficient over short intervals of a few minutes. Order imbalances are highly positively dependent over both short and long intervals and imbalances predict future returns over very short intervals. But order imbalances are known unambiguously only by the NYSE specialist.

Conditional on knowing the current order imbalance, returns are negatively serially dependent over intervals up to ten minutes. This conditional negative dependence in returns is consistent with NYSE specialists altering quotes away from fundamentals for the purpose of inventory risk control, while awaiting help from countervailing traders. Indeed, there is strong evidence that outside traders soon become aware of price-moving order imbalances and undertake countervailing trades. In no more than thirty minutes, order imbalances lose their predictive ability and returns are no longer negatively dependent.

24 Convergence to Efficiency, April 11, 2004

These results make one wonder about the existence of market anomalies and inefficiencies in general. Thus, the process of price formation permits no significant evidence of weak-form inefficiency at intervals of thirty minutes, but momentum at longer intervals of six months and beyond, as documented in the extensive literature on long-term anomalies.8 How markets transit from weak-form efficiency at very short horizons but predictability at long horizons appears to be a worthwhile area for future research.

8E.g., see Jegadeesh and Titman, 1993. Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998, 2001), and Hong and Stein (1999) attempt to explain momentum and other inefficiencies using models with irrational investors.

Convergence to Efficiency, April 11, 2004 25

References

Ball, Clifford A., and Tarun Chordia, 2001, True spreads and equilibrium prices, Journal of Finance 56, 1801-1835.

Barber, Brad M., and Terence Odean, 2000, Trading is hazardous to your health: The common stock investment performance of individual investors, Journal of Finance 55, 773-806.

Barberis, Nicholas, Andrei Shleifer, and Robert W. Vishny, 1998, A model of investor sentiment, Journal of Financial Economics 49 307-343.

Benartzi, Shlomo, and Richard Thaler, 2001, Naïve diversification strategies in retirement saving plans, American Economic Review 91, 79-98.

Bray, Margaret, 1981, Futures trading, rational expectations, and the efficient markets hypothesis, Econometrica 49, 575-596.

Brown, David P., and Robert E. Jennings, 1990, On technical analysis, Review of Financial Studies 2, 527-551.

Chakrabarti, Rajesh, and Richard Roll, 1999, Learning from others, reacting, and market quality, Journal of Financial Markets 2, 153-178.

Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2001, Order imbalance, liquidity and market returns, Journal of Financial Economics, 56, 3-28.

Copeland, Thomas E., 1976, A model of asset trading under the assumption of sequential information arrival, Journal of Finance 31, 1149-1168.

Cornell, Bradford and Richard Roll, 1981, Strategies for pairwise competitions in markets and organizations, Bell Journal of Economics 12, 201-213.

Cramér, Harald, 1954, Mathematical Methods of Statistics, (Princeton: Princeton University Press).

Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 1998, Investor psychology and security market under- and overreactions, Journal of Finance 53, 1839-1885.

Daniel, Kent, David Hirshleifer, and Avanidhar Subrahmanyam, 2001, Overconfidence, arbitrage, and equilibrium asset pricing, Journal of Finance 56, 921-965.

Epps, Thomas W., 1979, Comovements in stock prices in the very short run, Journal of the American Statistical Association 74, 291-298.

26 Convergence to Efficiency, April 11, 2004 Fama, Eugene F., 1970, Efficient capital markets: A review of theory and empirical work, Journal of Finance 25, 383-417.

Fama, Eugene F, and French, Kenneth R., 1992, The cross-section of expected stock returns, Journal of Finance 47, 427-465.

Garbade, Kenneth D., and Zvi Lieber, 1977, On the independence of transactions on the New York Stock Exchange, Journal of Banking and Finance 1, 151-172.

Grossman, Sanford J., 1976, On the efficiency of competitive stock-markets where traders have diverse information, Journal of Finance 31, 573-585.

Grossman, Sanford J. and Joseph E. Stiglitz, 1980, On the impossibility of informationally efficient markets, American Economic Review 70, 393-408.

Hillmer, S.C., and P. L. Yu, 1979, The market speed of adjustment to new information, Journal of Financial Economics 7, 321-345.

Hirshleifer, David, Avanidhar Subrahmanyam, and Sheridan Titman, 1994, Security analysis and trading patterns when some investors receive information before others, Journal of Finance 49, 1665-1698.

Hong, Harrison, and Jeremy C. Stein, 1999, A unified theory of underreaction, momentum trading, and overreaction in assets markets, Journal of Finance 54, 2143-2184.

Jegadeesh, Narasimhan, and Sheridan Titman, Returns to buying winners and selling losers: Implications for stock market efficiency, Journal of Finance 48, 65-91.

Jones, Charles, Gautam Kaul, and Marc Lipson, 1994, Transactions, volume, and volatility, Review of Financial Studies 7, 631-651.

Kendall, Maurice G., 1954, Note on bias in the estimation of autocorrelation, Biometrika 41, 403-404.

Kyle, Albert S., 1985, Continuous auctions and insider trading, Econometrica 53, 1315-1335.

Lee, Charles, and M. Ready, 1991, Inferring trade direction from intra-day data, Journal of Finance 46, 733-747.

Marriott, F. H. C., and J. A. Pope, 1954, Bias in the estimation of autocorrelations, Biometrika 41, 390-402.

Odean, Terrance, 1999, Do investors trade too much? American Economic Review 89, 1279- 1298.

Convergence to Efficiency, April 11, 2004 27 Patell, James M., and Mark A. Wolfson, 1984, The intra-day speed of adjustment of stock prices to earnings and dividend announcements, Journal of Financial Economics 13, 223-252.

Schwert, G. William, 2001, Anomalies and market efficiency, Chapter 17 in George Constantinides, Milton Harris, and René Stulz, eds., Handbook of the Economics of Finance, (North-Holland: Amsterdam).

28 Convergence to Efficiency, April 11, 2004 Appendix

Proof of Eq. (1): The expression we derive holds for multiple classes of agents with differential information sets and exponential utility within a dynamic model, not just the market makers we model. We begin by stating the following lemma, which is a standard result on multivariate normal random variables (see, for example, Brown and Jennings [1989]).

Lemma 1 Let Q(χ) be a quadratic function of the random vector χ: Q(χ) = C +B’ χ - χ ‘A χ, where χ ~ N(µ, Σ), and A is a square, symmetric matrix whose dimension corresponds to that of

χ . We then have

E[exp(Q(χ))] = | Σ|-1/2 |2A + Σ-1|-1/2 × exp(C + B’µ + µ’Aµ +(1/2)(B’ - 2µ’A’)( 2A + Σ-1)-1(B - 2Aµ) .

Let φij and xij denote the information set and demand, respectively, of an agent I at date j. The date 2 demand of the agent (from maximization of the mean-variance objective) is given by xi2 =[E(F|φi2) - P2]/[Rivar(F|φi2)] where Ri denotes the risk aversion coefficient of agent i. Let µ2 =E(F|φi2). Note that in period 1, the trader maximizes the derived expected utility of his time 2 wealth which is given by

2 E[[-exp{-Ri[B0 - xi1P1 + xi1P2 + [µ2 - P2] /(2Rivar(F|φi2))]}]|φi1]. (2)

Let р and µ denote the expectations of P2 and µ2, and Π the variance-covariance matrix of P2 and

µ2, conditional on φi1. Then, the expression within the exponential above (including terms from the normal density) can be written as - [(1/2)y’Gy + h’y + w , where y’ = [µ2 - µ, P2 -p], h = [-Rixi1 +(p - µ)/var(F|φi2),( µ-p)/ var(F|φi2)]

−1   s −1 − s −1  G = −1 + , ∏  −1 −1   − s s  w = Rixi1(P1 - p) + g, where s=var(F|φi1), and where g is an expression which does not involve xi1. From Lemma 1 and

Bray (1981, Appendix), (2) is given by -Det(Π)-1/2|Det(A)|-1/2exp[(1/2)h’G-1h-w]

Thus, the optimal xi1 solves

-1 [dh/dxi1]’G h-dw/dxi1= 0.

Substituting for h and w, we have

p − P1 µ − p G1 − G2 xi1 = + RiG1 Ri var(F | φi2 ) G1

-1 where G1 and G2 are the elements in the first row of the matrix G . It follows that the demand x1 is given by (1), with the S coefficients being the G1 and G2 coefficients above.

30 Convergence to Efficiency, April 11, 2004 Table 1 Sample Firms This table lists the ticker symbols and the market capitalization (as of the end of the relevant calendar year) for the largest 150 firms in each of the years 1996, 1999, and 2002, as measured by their market capitalization. The firms are divided into three groups of fifty firms each. 1996 1999 2002 Market Cap. Market Cap. Market Cap. Ticker Ticker Ticker ($bill) ($bill) ($bill) Largest 50 Firms GE 120.26 GE 334.237 GE 398.105 T 103.07 WMT 181.072 XON 268.833 XON 100.73 XON 177.784 C 259.710 KO 93.136 MRK 175.682 WMT 256.505 MRK 80.805 IBM 170.151 PFE 250.526 MO 75.335 KO 165.162 IBM 208.371 PG 56.965 PFE 162.224 AIG 207.431 JNJ 55.372 LU 144.965 JNJ 181.286 WMT 51.081 BMY 132.952 AOL 136.600 IBM 51.016 T 132.833 MRK 133.753 MOB 44.207 MO 130.255 SBC 131.672 PEP 44.025 PG 121.157 VZ 128.828 AIG 43.854 C 112.888 HD 119.534 BMY 43.330 JNJ 112.784 KO 117.251 BLS 43.216 SBC 104.891 PG 102.523 HWP 42.863 BAC 104.056 MO 99.343 GTE 42.498 AIG 101.452 BAC 99.041 PFE 40.078 LLY 97.736 BMY 98.678 GM 39.624 BLS 97.535 CVX 95.640 DD 38.803 HD 95.643 LLY 88.295 AN 35.519 BEL 83.823 ABT 86.616 SBC 34.914 SGP 81.196 PEP 85.188 CHV 34.161 FNM 75.776 AHP 80.948 ABT 32.902 ABT 74.363 FNM 79.580 AIT 32.610 AHP 74.303 WFC 73.691 MCD 31.420 CPQ 71.400 VIA 72.227 LLY 31.024 AOL 71.070 JPM 72.202 F 31.014 HWP 70.848 BLS 71.606 AHP 30.333 AIT 69.918 T 64.180 BEL 29.245 TWX 69.066 MDT 61.996 CCI 28.585 MOB 67.944 MWD 61.351 MMM 27.854 F 66.854 PHA 54.993 BA 26.894 WFC 64.585 SGP 52.429 BAC 23.957 GTE 62.732 TXN 48.502 NYN 23.220 WLA 61.760 AXP 47.618 G 23.131 DIS 61.524 MMM 46.348 KMB 23.045 FTU 60.250 ONE 45.610 EK 22.921 CMB 60.051 FRE 45.382 HD 22.768 PEP 59.973 GS 44.383 COL 22.592 ONE 59.794 DD 44.197 C 21.089 DD 59.741 MER 43.892 TX 20.733 AN 57.160 WB 42.677 SGP 20.124 CHV 54.112 DIS 42.240 AXP 20.042 G 53.039 USB 40.832 TRV 19.811 MCD 51.968 BUD 39.978 PNU 19.519 GM 46.836 HWP 39.848 NB 18.863 AXP 46.339 FBF 38.143 ALL 18.408 FRE 43.656 TGT 37.059 DOW 18.390 EMC 42.657 AWE 36.361 EMR 18.353 ATI 41.439 ADP 36.318

Convergence to Efficiency, April 11, 2004 31

Table 1, (Continued) 1996 1999 2002 Market Cap. Market Cap. Market Cap. Ticker Ticker Ticker ($bill) ($bill) ($bill) 50 Mid-cap Firms ARC 17.809 MWD 41.014 LOW 35.936 BUD 17.008 XRX 38.694 G 35.248 K 16.796 MOT 36.648 WAG 34.371 USW 16.773 MDT 36.350 MCD 34.026 SO 16.404 TXN 33.386 MOT 33.431 LMT 15.733 BA 32.597 EDS 32.496 SLE 15.492 USW 32.475 CL 31.862 S 15.204 GPS 32.042 BAX 31.661 FRE 15.094 ALL 31.589 KMB 31.404 JPM 15.050 BUD 31.363 BA 30.945 CPB 14.954 BK 30.617 DUK 30.471 FDC 14.927 SWY 29.628 DOW 30.466 XRX 14.776 KMB 29.533 CCU 30.435 ONE 14.722 AFS 29.348 UTX 30.282 TWX 14.675 WAG 29.206 AA 30.196 WMX 14.475 FON 28.981 BK 30.048 PAC 14.353 PNU 28.768 KRB 29.983 MTC 14.121 MTC 28.692 FDC 29.934 ATI 13.933 MMM 28.546 EMC 29.777 FON 13.821 UMG 28.524 MMC 29.510 CA 13.744 TX 28.330 CAH 29.154 UNP 13.568 CL 27.195 WM 28.642 ALD 13.436 WMI 26.780 DNA 28.601 WLA 13.149 EMR 26.505 HON 27.502 MDT 12.980 USB 25.763 F 27.368 CPQ 12.758 SLE 25.747 GM 26.997 GRN 12.714 FLT 25.398 HI 26.520 FCN 12.537 ALD 24.817 ALL 24.029 HNZ 12.245 EDS 24.730 EMR 24.018 PCG 11.890 UTX 24.559 COX 24.013 NOB 11.651 CPB 24.452 KSS 23.594 CAT 11.604 AUD 24.225 Q 23.526 UTX 11.576 NCC 23.979 P 22.976 RTN 11.452 DH 23.915 MET 22.961 UNH 11.421 MER 23.677 PCS 22.902 BAX 11.369 EK 23.324 UNH 22.065 BNI 11.066 DUK 23.189 LU 21.510 DNB 10.965 CBS 23.169 SCH 21.142 AUD 10.692 CA 22.943 SWY 20.954 JCP 10.652 WM 22.766 ITW 20.633 FLT 10.441 SCH 22.527 LMT 20.559 MD 10.306 CVS 21.448 WMI 20.033 NSC 10.300 ARC 21.000 CA 19.866 CL 10.227 HNZ 20.487 IP 19.447 WFC 10.145 SO 20.264 CD 19.247 CPC 10.002 FSR 20.237 AT 19.162 IP 9.876 DOW 20.128 THC 19.112 CAG 9.837 AGC 19.660 HCA 19.040 BK 9.751 HI 19.131 GIS 18.947 TXN 9.724 ENE 18.872 KFT 18.887

32 Convergence to Efficiency, April 11, 2004 Table 1, (Continued) 1996 1999 2002 Market Cap. Market Cap. Market Cap. Ticker Ticker Ticker ($bill) ($bill) ($bill) 50 Smaller Firms DUK 9.705 KRB 18.654 PRU 18.795 DEC 9.671 BAX 18.393 MU 18.578 CSX 9.603 JPM 18.367 UPS 18.302 ENE 9.593 COX 18.208 STI 18.109 ADM 9.543 GCI 18.163 CAT 17.935 FTU 9.479 LOW 18.054 FON 17.839 AA 9.370 PBI 18.011 GCI 17.806 TXU 9.259 MEL 17.953 NCC 17.720 LTR 9.235 WB 17.728 COC 17.701 DE 9.233 LMT 16.633 MEL 17.600 GIS 9.159 GDT 16.570 SYY 17.476 PPG 8.969 CAT 16.500 SLE 17.413 MER 8.962 CD 16.475 SO 17.341 UMG 8.957 AT 16.414 STT 17.000 P 8.943 S 16.282 KR 16.705 WY 8.724 PNC 16.235 OMC 16.606 GCI 8.605 BNI 16.067 CPQ 16.592 KEY 8.582 STI 16.032 BBT 16.485 FPL 8.574 CI 15.964 HDI 16.441 CB 8.438 ABS 15.641 GD 16.070 RN 8.385 KR 15.501 ADI 16.059 AMP 8.323 COL 15.457 PNC 15.965 ABS 8.313 CAG 15.414 LEH 15.919 MU 8.180 CAH 15.230 D 15.792 AMB 8.076 BFO 15.217 BBY 15.705 BSX 7.975 MMC 15.036 FDX 15.467 DWD 7.938 ITW 14.504 S 15.369 CI 7.841 CCU 14.358 EXC 15.364 WB 7.792 FDC 14.109 GDT 15.175 STI 7.751 KEY 14.076 HIG 14.959 MEL 7.595 CCE 14.011 FRX 14.584 AEP 7.545 K 13.831 HNZ 14.394 PEG 7.494 IP 13.771 UNP 14.159 ED 7.460 AA 13.688 LUV 14.144 PNC 7.393 MAY 13.501 APC 14.142 WAG 7.353 WMB 13.341 AEP 14.027 GLW 7.336 TXU 13.181 GMH 13.543 UCL 7.198 FDX 13.160 CI 13.400 D 7.157 HAL 13.025 SLM 13.172 RPR 7.150 EQ 12.913 WMB 13.152 AGC 7.142 RAD 12.859 A 13.143 PBI 7.125 HIG 12.510 AFL 12.825 UCM 7.035 MU 12.464 ABS 12.795 CNA 7.014 ED 12.330 CAG 12.771 ITW 6.936 PGR 12.278 RTN 12.693 GT 6.917 CLX 12.093 TXU 12.500 OXY 6.807 RX 12.082 JHF 12.335 WX 6.800 PCG 12.049 BHI 12.249 MAT 6.791 TXT 12.031 CPB 12.237 PE 6.687 NSC 12.016 K 12.235

Convergence to Efficiency, April 11, 2004 33 Table 2 Correlation Coefficients at a Daily Horizon for Returns and Order Imbalances

For stocks listed in Table 1, trade returns are computed from the last transaction price of each day and midpoint returns are computed from the average of the bid-ask quotes associated with the last transaction of each day. Trade returns are from CRSP. Bid-Ask quotes and order imbalances (OIB) are from the NYSE TAQ data base. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same day as the return; OIBSh is the number of buyer-initiated shares purchased less the number of seller-initiated shares sold that day; OIB$ is the total dollars paid by buyer-initiators less the total dollars received by seller-initiators that day. The product-moment correlation coefficient is reported along with a t-statistic computed from the cross-sectional distribution of correlation coefficients.

Trade Midpoint Trade Midpoint Trade Midpoint OIB#t OIBSht OIB$t OIB#t OIBSht OIB$t OIB#t OIBSht OIB$t Returnt Returnt Returnt Returnt Returnt Returnt 1996 1999 2002 Large Stocks 0.005 0.011 0.006 0.013 -0.028 -0.028 Return 9 t-1 (0.48) (1.03) (0.32) (0.70) (-2.98) (-2.92) 0.270 0.271 0.117 0.116 0.187 0.186 OIB# t (13.45) (13.26) (4.16) (4.08) (7.00) (6.93) 0.071 0.071 0.436 0.046 0.048 0.440 0.046 0.046 0.317 OIB# t-1 (7.99) (8.06) (15.75) (4.28) (4.58) (18.14) (3.97) (3.99) (13.31) 0.525 0.534 0.286 0.550 0.556 0.197 0.409 0.410 0.451 OIBSh t (39.22) (40.35) (13.43) (36.89) (36.87) (7.49) (22.27) (22.17) (19.09) 0.001 -0.003 -0.084 0.149 -0.008 -0.007 -0.088 0.217 0.014 0.013 0.045 0.168 OIBSh t-1 (0.13) (-0.31) (-4.07) (9.85) (-0.65) (-0.56) (-5.27) (8.67) (1.22) (1.13) (2.85) (11.08) 0.524 0.532 0.277 0.990 0.547 0.553 0.199 0.986 0.401 0.402 0.449 0.983 OIB$ t (37.91) (38.74) (12.70) (442.0) (37.24) (37.16) (7.61) (378.0) (24.06) (23.91) (19.58) (407.8) -0.003 -0.008 -0.091 0.145 0.149 -0.012 -0.011 -0.083 0.209 0.219 0.001 -0.000 0.039 0.152 0.159 OIB$ t-1 (-0.28) (-0.72) (-4.60) (9.83) (9.77) (-0.96) (-0.90) (-4.65) (8.45) (8.65) (0.07) (-0.02) (2.68) (10.70) (10.92)

9 Trade (Midpoint) Returnt-1 in the Trade Returnt (Midpoint Returnt) column.

34 Convergence to Efficiency, April 11, 2004 Table 2, (Continued)

Trade Midpoint Trade Midpoint Trade Midpoint OIB#t OIBSht OIB$t OIB#t OIBSht OIB$t OIB#t OIBSht OIB$t Returnt Returnt Returnt Returnt Returnt Returnt 1996 1999 2002 Mid-cap stocks Return 0.017 0.020 0.013 0.023 -0.042 -0.038 t- 1 (1.62) (1.88) (1.06) (1.87) (-4.51) (-4.11) 0.317 0.315 0.238 0.236 0.237 0.234 OIB# t (10.74) (10.74) (7.60) (7.43) (8.00) (7.78) 0.042 0.043 0.302 0.021 0.024 0.373 0.007 0.006 0.234 OIB# t-1 (4.36) (4.47) (10.77) (2.13) (2.51) (17.94) (0.65) (0.54) (9.14) 0.434 0.440 0.307 0.455 0.459 0.297 0.315 0.313 0.519 OIBSh t (21.97) (22.23) (10.14) (22.13) (22.00) (9.08) (15.94) (15.69) (32.37) OIBSh 0.006 0.001 -0.058 0.134 -0.005 -0.001 0.014 0.205 0.009 0.010 0.109 0.188 t- 1 (0.62) (0.07) (-3.22) (8.11) (-0.46) (-0.09) (0.74) (15.15) (0.85) (0.91) (5.77) (8.55) 0.430 0.436 0.309 0.987 0.451 0.455 0.305 0.979 0.305 0.303 0.505 0.950 OIB$ t (22.36) (22.53) (10.71) (237.6) (21.72) (21.59) (9.54) (191.6) (16.07) (15.91) (31.51) (104.6) 0.006 0.001 -0.055 0.133 0.137 -0.009 -0.006 0.023 0.197 0.210 -0.003 -0.002 0.098 0.169 0.199 OIB$ t-1 (0.59) (0.07) (-3.18) (8.30) (8.48) (-0.90) (-0.52) (1.13) (15.81) (15.40) (-0.37) (-0.26) (5.52) (8.41) (8.88) Smaller Stocks Return -0.015 -0.010 -0.019 -0.004 -0.020 -0.017 t- 1 (-1.38) (-0.96) (-1.69) (-0.41) (-1.65) (-1.36) 0.357 0.354 0.286 0.285 0.271 0.271 OIB# t (16.60) (16.43) (10.30) (10.17) (8.92) (8.81) 0.052 0.054 0.271 -0.009 -0.003 0.304 0.013 0.014 0.195 OIB# t-1 (4.81) (4.93) (9.75) (-0.83) (-0.31) (14.46) (1.43) (1.42) (10.37) 0.402 0.406 0.323 0.386 0.392 0.338 0.257 0.256 0.538 OIBSh t (20.34) (20.41) (16.38) (17.82) (17.95) (12.95) (11.62) (11.54) (28.81) OIBSh -0.002 -0.004 -0.022 0.095 -0.035 -0.032 0.017 0.192 -0.006 -0.005 0.110 0.182 t- 1 (-0.23) (-0.43) (-1.50) (6.77) (-3.52) (-3.38) (0.78) (10.67) (-0.73) (-0.53) (6.30) (10.15) 0.402 0.406 0.323 0.995 0.378 0.384 0.331 0.980 0.258 0.257 0.511 0.962 OIB$ t (21.07) (21.17) (16.24) (903.5) (18.26) (18.34) (12.78) (284.3) (13.38) (13.30) (30.63) (91.53) -0.001 -0.004 -0.025 0.094 0.097 -0.039 -0.037 0.016 0.187 0.197 -0.017 -0.016 0.090 0.162 0.183 OIB$ t-1 (-0.15) (-0.41) (-1.58) (6.59) (6.67) (-3.96) (-3.88) (0.75) (10.74) (10.40) (-2.00) (-1.80) (6.46) (10.20) (11.02)

Convergence to Efficiency, April 11, 2004 35 Table 3

Univariate Regressions for Return Intervals from Five to Sixty Minutes

Daily returns and order imbalances are obtained from the NYSE TAQ data base for the 150 large stocks listed in Table 1. The return is computed from the midpoint of the bid-ask spread associated with the transaction nearest the end of an intra-day time interval of fixed length. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same time interval as the return. OIB$ is the total dollar amount expended by buyer-initiators less the total dollar amount received by seller-initiators during that interval. The first interval of each day is excluded and all other interval observations during each calendar year, (either 1996, 1999 or 2002), are included in the same regression. A separate regression is estimated for each individual stock. In each case, the dependent variable is the next period’s mid-point return. The first number in each cell is the cross-sectional mean of the estimated regression coefficient. The second number (in parentheses) is the average t-statistic from the individual regressions. The third number (also in parentheses) is a t-statistic computed from the cross-sectional distribution of the estimated coefficients adjusting for cross-correlation in the residuals. The fourth number is the cross-sectional average adjusted R-square in percent. To adjust the units for presentation, the coefficients for OIB# and OIB$ have been multiplied by 105.

36 Convergence to Efficiency, April 11, 2004 Table 3, (Continued)

Return interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Large stocks, 1996 -0.001 -0.040 -0.041 1.87 0.038 Midpoint (-0.23) (-3.90) (-3.27) (0.99) (1.30) Returnt-1 (-0.06) (-2.09) (-1.98) (0.86 (1.39) 0.18 0.29 0.32) 0.170. 0.30 32.89 195.30 0.584 0.873 0.737 (14.35) (19.66) (1.55) (2.15) (1.52) OIB# t-1 (3.71) (4.43) (1.25) (2.09) (1.46) 1.23 4.76 0.12 0.23 0.29 114.76 49.33 36.60 42.33 45.01 (11.92) (3.65) (2.26) (2.08) (1.51) OIB$ t-1 (4.31) (2.58) (1.84) (2.72) (1.89) 0.81 0.19 0.14 0.18 0.25 Large stocks, 1999 -0.000 0.004 0.009 0.019 0.038 Midpoint (-0.01) (0.38) (0.72) (1.06) (1.34) Returnt-1 (-0.01) (0.25) (0.63) (1.37) (1.79) 0.13 0.37 0.08 0.09 0.24 0.806 0.431 0.403 0.353 0.557 (4.25) (1.84) (1.75) (1.24) (1.31) OIB# t-1 (1.73) (1.28) (1.46) (1.33) (1.77) 0.17 0.07 0.08 0.11 0.19 2.32 1.31 1.30 2.09 2.65 (3.05) (1.45) (1.32) (1.41) (1.18) OIB$ t-1 (1.95) (1.50) (1.50) (1.59) (1.59) 0.09 0.06 0.63 0.14 0.17 Large Stocks, 2002 -0.031 -0.033 -0.025 -0.009 0.000 Midpoint (-4.29) (-3.27) (-1.95) (-0.51) (0.02) Returnt-1 (-1.49) (-1.19) (-1.19) (-0.43) (0.00) 0.23 0.18 0.18 0.00 0.12 0.660 0.179 0.112 0.031 0.070 (3.58) (0.57) (0.34) (0.18) (0.23) OIB# t-1 (1.16) (0.46) (0.28) (0.08) (0.17) 0.15 0.06 0.07 0.05 0.05 3.76 1.78 2.44 0.616 0.798 (2.77) (0.95 (0.68) (0.34) (0.38) OIB$ t-1 (1.37) (0.91) (0.70) (0.28) (0.27 0.07 0.03 0.03 0.04 0.04

Convergence to Efficiency, April 11, 2004 37 Table 3, (Continued)

Return interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Mid-cap stocks, 1996 0.015 -0.023 -0.018 0.007 0.005 Midpoint (1.42) (-1.35) (-1.44) (0.33) (0.17) Returnt-1 (0.61) (-0.50) (-0.71) (0.41) (0.17) 0.45 0.34 0.34 0.13 0.32 80.84 37.33 23.29 15.92 0.764 (19.89) (6.83) (3.72 (2.00) (0.77) OIB# t-1 (6.35) (4.31) (3.49) (2.77) (0.88) 2.85 0.78 0.37 0.23 0.23 206.34 122.22 88.64 56.71 -0.472 (12.35) (5.17) (3.20) (1.66) (1.01) OIB$ t-1 (5.27) (4.07) (3.64) (2.78) (-0.11) 1.08 0.41 0.24 0.15 0.03 Mid-cap stocks, 1999 0.023 0.012 0.008 0.000 0.014 Midpoint (3.16) (1.21) (0.63) (0.01) (0.50 Returnt-1 (1.75) (0.93) (0.53) (0.02) (0.71) 0.17 0.13 0.14 0.12 0.17 2.75 1.33 0.831 0.348 0.404 (9.39) (3.52) (1.93) (0.66) (0.58) OIB# t-1 (4.43) (2.93) (1.91) (0.84) (0.77) 0.55 0.18 0.12 0.07 0.14 7.09 3.86 3.23 1.36 1.90 (5.47) (2.14) (1.52) (0.52) (0.60) OIB$ t-1 (4.06) (2.94) (2.45) (1.01) (0.94) 0.19 0.07 0.06 0.03 0.08 Mid-cap stocks, 2002 0.001 -0.013 -0.011 0.000 -0.003 Midpoint (0.10) (-1.28) (-0.91) (0.01) (-0.13) Returnt-1 (0.04) (-0.68) (-0.62) (0.01) (0.11) 0.14 0.18 0.15 0.11 0.15 1.60 0.539 0.458 0.219 0.216 (7.56) (1.91) (1.33) (0.43) (0.43) OIB# t-1 (3.26) (1.43) (1.17) (0.53 (0.52 0.36 0.10 0.10 0.07 0.08 77.61 3.21 3.49 2.21 -1.15 (4.19) (1.31) (1.13) (0.53) (0.10) OIB$ t-1 (3.19) (1.59) (1.43) (0.72) (-0.05) 0.12 0.05 0.06 (0.05 0.05

38 Convergence to Efficiency, April 11, 2004 Table 3, (Continued)

Return interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Small stocks, 1996 -0.022 -0.025 -0.024 0.002 0.019 Midpoint (-2.60) (-2.33) (-1.94) (-0.13) (0.33) Returnt-1 (-0.79) (-0.91) (-0.82) (0.06) (0.54) 0.72 0.52 0.51 0.35 0.23 118.33 62.08 45.82 28.20 15.00 (19.91) (7.82) (4.82) (2.67) (1.24 OIB# t-1 (6.47) (4.92) (3.93) (2.78) (1.03) 3.74 1.29 0.82 0.42 0.18 320.59 186.82 146.97 114.13 -4.72 (11.48) (5.10) (3.23) (1.91) (1.01) OIB$ t-1 (4.74) (3.11) (3.24) (2.38) (-0.11) 1.22 0.55 0.27 0.14 0.33 Small stocks, 1999 0.025 0.015 0.009 0.009 0.013 Midpoint (3.35) (1.39) (0.64) (0.41 (0.43) Returnt-1 (1.59) (1.00) (0.60) 0.45 (0.77) 0.29 0.21 0.16 0.22 0.16 49.49 2.51 1.59 0.782 0.627 (12.00) (4.39) (2.41) (0.92) (0.55) OIB# t-1 (4.92) (3.64) (2.99) (1.26) (1.08) 0.97 0.32 0.16 0.11 0.13 131.10 7.30 5/01 3.84 3.48 (6.18) (2.45) (1.47) (0.82) (0.58) OIB$ t-1 (4.08) (3.13) (2.93) (1.61) (1.13) 0.26 0.09 0.05 0.06 0.09 Small stocks, 2002 -0.009 -0.023 -0.012 -0.003 0.006 Midpoint (-0.99) (-2.11) (-0.81) (-0.09 (0.21) Returnt-1 (-0.41) (-0.98) (-0.46) (-0.15) (0.27) 0.25 0.29 0.29 0.16 0.09 1.59 0.527 0.372 0.217 0.260 (6.63) (1.54) (0.98) (0.50) (0.52) OIB# t-1 (2.71) (1.26) (1.08) (0.55) (0.75) 0.33 0.09 0.07 0.09 0.07 9.81 4.87 38.38 27.50 2.65 (3.92) (1.29) (0.99) (0.47) (0.34) OIB$ t-1 (2.16) (1.40) (1.16) (0.60) (0.52) 0.13 0.04 0.06 0.06 0.08

Convergence to Efficiency, April 11, 2004 39

Table 4 Univariate Regressions for Order Imbalance Intervals from Five to Sixty Minutes OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same time interval as the return. OIB$ is the total dollar amount expended by buyer-initiators less the total dollar amount received by seller-initiators during that interval. The first interval of each day is excluded and all other interval observations during each calendar year, (either 1996, 1999 or 2002), are included in the same regression. The dependent variable is order imbalance in numbers of transactions for OIB#t-1 and in dollars for OIB$t-1 A separate regression is estimated for each individual stock. The first number in each cell is the cross-sectional mean of the estimated regression coefficient. The second number (in parentheses) is the average t-statistic from the individual regressions. The third number (also in parentheses) is a t-statistic computed from the cross-sectional distribution of the estimated coefficients adjusting for cross-correlation in the residuals. The fourth number is the cross-sectional average adjusted R-square in percent.

Imbalance interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Large stocks, 1996 0.223 0.221 0.237 0.301 0.372 (32.82 (23.18) (20.45) (18.61) (15.87) OIB# t-1 (0.974) (5.95) (5.22) (5.65) (5.75) 6.00 6.56 7.77 12.01 17.38 0.077 0.068 0.070 0.086 0.113 (10.48) (6.55) (5.42) (4.62) (4.01) OIB$ t-1 (11.22) (8.45) (7.77) (7.21) (5.43) 0.66 0.55 0.58 0.89 1.68 Large stocks, 1999 0.204 0.260 0.286 0.337 0.425 (31.59 (28.92) (26.16) (21.98) (18.82) OIB# t-1 (10.61) (9.62) (9.56) (9.67) (7.37) 6.00 9.72 11.63 15.92 22.74 0.097 0.118 0.136 0.155 0.192 (13.57) (11.62) (10.86) (8.61) (7.01) OIB$ t-1 (9.67) (9.76) (10.34) (9.86) (11.56) 1.19 1.72 2.22 2.88 4.39 Large stocks, 2002 0.153 0.195 0.230 0.291 0.359 (21.68) (19.66) (18.96) (17.12) (6.23) OIB# t-1 (5.81) (5.45) (5.45) (5.78) (6.34) 2.90 4.76 6.52 10.32 15.02 0.072 0.091 0.115 0.143 0.182 (9.56) (8.53) (8.69) (7.42) (6.19) OIB$ t-1 (10.56) (7.36) (7.95) (7.25) (6.65) 0.55 0.94 1.44 2.22 3.47

40 Convergence to Efficiency, April 11, 2004

Table 4, (Continued)

Imbalance interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Mid-cap stocks, 1996 0.218 0.202 0.204 0.236 0.265 (28.49) (18.81) (15.71) (12.95) (9.98) OIB# t-1 (13.23) (6.13) (4.88) (4.68) (4.07) 5.19 5.15 5.76 8.02 10.43 0.069 0.062 0.068 0.068 0.091 (8.54) (4.68) (4.71) (3.23) (2.83) OIB$ t-1 (14.16) (7.79) (8.96) (6.03) (6.50) 0.53 0.52 0.55 0.69 1.10 Mid-cap stocks, 1999 0.171 0.203 0.222 0.268 0.338 (24.76) (20.79) (18.56) (15.69) (13.53) OIB# t-1 (9.28) (7.34) (7.25) (6.72) (7.12) 3.51 5.06 6.07 8.88 13.66 0.063 0.078 0.093 0.113 0.149 (8.70) (7.60) (7.35 (6.08) (5.45) OIB$ t-1 (12.53) (11.51) (9.60) (8.66) (8.19) 0.47 0.74 1.14 1.69 3.00 Mid-cap stocks, 2002 0.173 0.199 0.229 0.277 0.339 (25.03) (20.33) (19.10) (16.31) (13.45) OIB# t-1 (6.36) (5.90) (6.11) (6.27) (6.67) 3.77 4.86 6.29 9.12 13.10 0.069 0.089 0.104 0.133 0.168 (9.47) (8.50) (8.07) (7.12) (5.39) OIB$ t-1 (13.16) (9.59) (9.38) (7.44) (9.56) 0.53 0.93 1.26 2.16 2.90

Convergence to Efficiency, April 11, 2004 41 Table 4, (Continued)

Imbalance interval (minutes) Explanatory Five Ten Fifteen Thirty Sixty variable Small stocks, 1996 0.218 0.204 0.208 0.230 0.263 (25.28) (16.48) (13.61) (10.80) (8.32) OIB# t-1 (19.94) (10.56) (8.23) (7.03) (5.91) 5.11 4.80 5.28 6.75 9.00 0.058 0.062 0.058 0.061 0.062 (6.19) (4.68) (3.68) (2.85) (1.74) OIB$ t-1 (10.23) (7.79) (5.74) (3.02) (3.20) 0.42 0.52 0.48 0.74 0.76 Small stocks, 1999 0.168 0.203 0.205 0.242 0.297 (23.91) (20.79) (17.08) (14.17) (11.73) OIB# t-1 (8.73) (7.34) (5.89) (5.42) (5.90) 3.42 5.06 5.62 7.92 11.32 0.054 0.070 0.075 0.099 0.121 (7.39) (6.63) (5.81) (5.29) (4.35) OIB$ t-1 (10.07) (9.19) (7.42) (7.44) (6.51) 0.40 0.66 0.86 1.47 2.42 Small stocks, 2002 0.167 0.201 0.229 0.275 0.330 (22.36) (19.07) (17.72) (14.98) (11.98) OIB# t-1 (8.87) (7.45) (7.53) (7.34) (8.64) 3.02 4.50 5.80 8.56 11.71 0.064 0.082 0.096 0.123 0.168 (16.18) (7.16) (7.03) (6.02) (5.39) OIB$ t-1 (14.28) (10.24) (10.89) (11.04) (9.56) 0.45 0.67 0.98 1.55 2.90

42 Convergence to Efficiency, April 11, 2004 Table 5 Cross-Autocorrelation Coefficients between Buying and Selling over Intra-Day Horizons Buys and sells are estimated from NYSE TAQ data for stocks listed in Table 1. The first interval of each day is excluded. The reported product-moment autocorrelation coefficient is between orders on one side of the market during a trading interval and orders on the opposite side of the market during the subsequent interval. In other words, buy orders are correlated with subsequent sell orders, and vice versa. The t-statistic (in parentheses) is computed from the cross-sectional distribution of correlation coefficients. A * next to a coefficient indicates that it is significantly larger than the corresponding coefficient for the five minute interval.

Panel A: Number of Orders Initial Five Ten Fifteen Thirty Sixty Order type Trading Interval (minutes) 0.203 0.280 0.328 0.390 0.392 Buy (14.41) (18.79) (21.76) (26.53) (24.81) 1996 0.198 0.287 0.343 0.408 0.415 Sell (13.59) (18.58) (21.99) (25.99) (24.05) 0.364 0.460 0.509 0.560 0.533 Buy (25.26) (31.09) (35.15) (40.40) (32.03) Large Cap 1999 0.371 0.466 0.512 0.568 0.544 Sell (25.12) (30.74) (34.89) (41.79) (32.25) 0.339 0.434 0.485 0.540 0.511 Buy (22.25) (27.14) (31.28) (37.03) (31.80) 2002 0.337 0.436 0.490 0.553 0.536 Sell (21.99) (26.75) (30.33) (37.20) (32.53) 0.131 0.206 0.251 0.317 0.349 Buy (7.01) (10.03) (11.71) (14.16) (15.02) 1996 0.121 0.199 0.248 0.318 0.347 Sell (6.32) (9.58) (11.68) (14.71) (15.89) 0.298 0.393 0.445 0.507 0.494 Buy (16.41) (19.80) (22.47) (25.92) (22.44) Mid-cap 1999 0.294 0.389 0.441 0.509 0.504 Sell (16.22) (20.29) (22.92) (27.49) (24.82) 0.319 0.415 0.462 0.523 0.511 Buy (17.65) (22.28) (24.80) (28.48) (26.72) 2002 0.321 0.419 0.473 0.538 0.533 Sell (17.76) (21.90) (24.73) (29.74) (28.91) 0.076 0.130 0.174 0.243 0.290 Buy (4.44) (6.28) (8.63) (11.40) (12.31) 1996 0.058 0.130 0.177 0.241 0.303 Sell (3.16) (6.66) (8.70) (11.24) (13.70) 0.231 0.316 0.363 0.423 0.408 Buy (11.64) (14.22) (16.27) (18.73) (16.71) Smaller Cap 1999 0.227 0.312 0.360 0.426 0.426 Sell (10.88) (13.88) (15.81) (19.48) (18.43) 0.312 0.404 0.450 0.505 0.481 Buy (16.01) (19.94) (22.65) (25.93) (22.90) 2002 0.309 0.404 0.455 0.512 0.496 Sell (16.01) (19.97) (22.62) (25.84) (23.65)

Convergence to Efficiency, April 11, 2004 43 Table 5, (Continued)

Panel B: Dollar Value of Orders

Initial Five Ten Fifteen Thirty Sixty Order type Trading Interval (minutes) 0.090 0.148 0.182 0.242 0.286 Buy (9.92) (11.98) (13.20) (16.47) (18.47) 1996 0.081 0.139 0.181 0.239 0.274 Sell (8.30) (10.93) (13.00) (15.32) (17.37) 0.204 0.292 0.336 0.402 0.401 Buy (13.65) (17.16) (19.11) (23.46) (24.64) Large Cap 1999 0.200 0.286 0.333 0.394 0.396 Sell (13.65) (16.58) (18.64) (21.86) (25.45) 0.241 0.342 0.395 0.452 0.434 Buy (22.00) (26.49) (29.02) (32.51) (28.91) 2002 0.239 0.342 0.394 0.459 0.447 Sell (21.74) (26.17) (29.13) (31.96) (29.05) 0.064 0.108 0.140 0.194 0.236 Buy (6.97) (8.24) (9.20) (10.41) (11.09) 1996 0.055 0.099 0.130 0.190 0.230 Sell (5.98) (7.55) (8.43) (10.20) (11.71) 0.136 0.209 0.251 0.320 0.359 Buy (10.10) (11.89) (13.35) (15.93) (18.31) Mid-cap 1999 0.130 0.203 0.245 0.311 0.344 Sell (9.66) (12.08) (13.38) (15.76) (17.77) 0.222 0.322 0.251 0.320 0.359 Buy (10.10) (11.89) (13.35) (15.93) (18.31) 2002 0.221 0.323 0.375 0.449 0.471 Sell (20.32) (22.21) (24.07) (25.42) (24.09) 0.047 0.075 0.102 0.138 0.172 Buy (4.38) (5.64) (6.99) (8.87) (9.01) 1996 0.042 0.079 0.104 0.135 0.178 Sell (3.79) (5.31) (6.04) (8.16) (8.39) 0.106 0.164 0.203 0.258 0.305 Buy (8.85) (10.39) (11.62) (13.82) (16.93) Smaller Cap 1999 0.100 0.158 0.197 0.258 0.296 Sell (8.93) (10.44) (11.68) (13.71) (15.65) 0.204 0.292 0.341 0.405 0.418 Buy (15.18) (18.42) (20.24) (23.22) (22.67) 2002 0.203 0.291 0.341 0.403 0.421 Sell (15.04) (16.78) (18.92) (21.83) (22.67)

44 Convergence to Efficiency, April 11, 2004

Table 6

Multiple Regressions of Returns on Lagged Returns and Two Different Measures of Lagged Order Imbalance for Return Intervals from Five to Sixty Minutes

Daily returns and order imbalances are obtained from the NYSE TAQ data base for the 150 large stocks listed in Table 1. The return is computed from the midpoint of the bid-ask spread associated with the transaction nearest the end of an intra-day time interval of fixed length. OIB# is the number of buyer-initiated less the number of seller-initiated trades during the same time interval as the return. OIB$ is the total dollar amount expended by buyer-initiators less the total dollar amount received by seller-initiators during that interval. The first interval of each day is excluded and all other interval observations during each calendar year, (either 1996 or 1999 or 2002), are included in the same regression. A separate regression is estimated for each individual stock. The first number in each cell is the cross-sectional mean of the estimated regression coefficient. The first number in each cell is the cross-sectional mean of the estimated regression coefficient. The second number (in parentheses) is the average t-statistic from the individual regressions. The third number (also in parentheses) is a t-statistic computed from the cross-sectional distribution of the estimated coefficients adjusting for cross-correlation in the residuals. The R2 is the cross-sectional average adjusted R-square in percent. To adjust the units for presentation, the coefficients for OIB# and OIB$ have been multiplied by 105.

Convergence to Efficiency, April 11, 2004 45

Table 6 (Continued)

Explanatory Return Interval (minutes) Variable Five Ten Fifteen Thirty Sixty Dependent Variable is the Midpoint Returnt, Large Stocks, 1996 -0.050 -0.039 -0.068 -0.069 -0.063 -0.068 0.001 -0.000 0.024 0.021 Midpoint (-6.39) (-5.04) (-5.94) (-6.06) (-4.45) (-4.79) (0.06) (-0.04) (0.75) (0.59) Return t-1 (-2.70) (-1.67) (-3.51) (-3.30) (-3.00) (-3.16) (0.05) (-0.01) (0.90) (0.69) 38.65 1.87 1.33 0.921 0.555 OIB#t-1 (15.35) (5.46) (3.25) (1.91) (1.07) (3.63) (2.75) (2.45) (1.99) (1.25) 134.16 8.53 7.28 4.43 3.70 OIB$t-1 (12.98) (5.93) (4.19) (1.81) (0.95) (4.09) (3.49) (3.09) (2.39) (1.37) R2 1.58 1.16 0.71 0.71 0.57 0.64 0.35 0.31 0.45 0.43 Dependent Variable is the Midpoint Returnt, Mid-cap Stocks, 1996 -0.060 -0.016 -0.061 -0.038 -0.053 -0.040 -0.015 -0.008 -0.008 -0.012 Midpoint (-7.25) (-2.37) (-5.02) (-3.41) (-3.53) (-2.89) (-0.69) (-0.40) (0.76) (-0.34) Return t-1 (-3.23) (-0.61) (-2.83) (-1.41) (-2.07) (-1.44) (-0.73) (-0.36) (-0.25) (-0.36) 9.11 4.87 3.29 1.90 0.954 OIB#t-1 (20.47) (7.98) (4.65) (2.01) (0.76) (6.53) (4.92) (3.89) (2.57) (1.20) 218.85 151.21 121.03 6.69 6.24 OIB$t-1 (12.45) (6.02) (4.03) (1.65) (1.04) (4.93) (3.87) (3.45) (2.41) (1.78) R2 3.36 1.53 1.30 0.88 0.88 0.39 0.31 0.47 0.47 Dependent Variable is the Midpoint Returnt, Small Stocks, 1996 -0.096 -0.049 -0.081 -0.049 -0.072 -0.046 -0.030 -0.015 -0.000 0.006 Midpoint (-10.04) (-5.62) (-6.03) (-4.08) (-4.38) (-3.19) (-1.42) (-0.90) (-0.26) (-0.10) Return t-1 (-4.49) (-1.83) (-3.38) (-1.70) (-2.66) (-4.43) (-0.99) (-0.47) (-0.01) (0.16) 137.55 7.79 5.88 3.17 14.28 OIB#t-1 (21.51) (9.13) (5.87 (2.82) (1.12) (5.85) (5.43) (4.65) (2.11) (0.66) 350.44 219.84 182.88 9.37 -5.41 OIB$t-1 (12.41) (5.97) (4.02) (2.02) (0.92) (5.02) (3.28) (3.38) (1.00) (-0.12) R2 5.02 2.07 2.18 1.23 1.57 0.93 0.97 0.61 0.60 0.60

46 Convergence to Efficiency, April 11, 2004

Table 6 (Continued)

Explanatory Return Interval (minutes) Variable Five Ten Fifteen Thirty Sixty Dependent Variable is the Midpoint Returnt, Large Stocks, 1999 -0.012 -0.010 -0.004 -0.003 0.001 0.002 0.013 0.010 0.027 0.026 Midpoint (-1.57) (-1.21) (-0.79) (-0.25) (0.11) (0.13) (0.66) (0.48) (0.89) (0.85) Return t-1 (-0.73) (-0.55) (-0.27) (-0.20) (0.05) (0.11) (0.91) (0.67) (1.23) (1.10) 0.906 0.490 0.445 0.285 0.398 OIB#t-1 (4.44) (1.88) (1.70) (0.97) (0.91) (1.76) (1.47) (1.55) (1.16) (1.34) 2.66 1.47 1.32 1.78 17.29 OIB$t-1 (9.22) (1.44) (1.14) (1.07) (0.67) (2.19) (1.72) (1.36) (1.38) (0.94) R2 0.29 0.22 0.15 0.13 0.15 0.12 0.31 0.17 0.34 0.32 Dependent Variable is the Midpoint Returnt, Mid-cap Stocks, 1999 -0.007 0.012 -0.004 0.006 -0.004 0.001 -0.007 -0.004 0.006 0.008 Midpoint (-0.89) (1.57) (-0.42) (0.53) (-0.33) (0.08) (-0.36) (-0.22) (0.22) (0.26) Return t-1 (-0.55) (0.88) (-0.33) (0.44) (-0.30) (0.10) (-0.46) (-0.24) (0.30) (0.39) 2.88 1.39 0.940 0.534 0.410 OIB#t-1 (8.84) (3.26) (1.84) (0.76) (0.45) (4.33) (3.22) (2.43) (1.31) (0.70) 6.66 3.52 3.18 1.72 1.71 OIB$t-1 (4.77) (1.77) (1.32) (0.50) (0.41) (3.77) (2.76) (2.86) (1.33) (0.84) R2 0.66 0.32 0.28 0.18 0.23 0.18 0.18 0.13 0.28 0.22 Dependent Variable is the Midpoint Returnt, Small Stocks, 1999 -0.014 0.014 -0.007 0.008 -0.008 0.003 -0.001 0.003 0.006 0.008 Midpoint (-1.77) (1.77) (-0.68) (0.71) (-0.56) (0.18) (-0.06) (0.10) (0.18) (0.23) Return t-1 (0.94) (0.88) (-0.49) (0.53) (-0.53) (0.20) (-0.06) (0.15) (0.34) (0.45) 5.29 2.67 18.09 0.870 0.586 OIB#t-1 (11.52) (4.11) (2.38) (0.84) (0.42) (4.83) (3.72) (3.39) (1.61) (1.11) 12.25 66.89 4.82 3.62 2.42 OIB$t-1 (15.72) (2.08) (1.31) (0.69) (0.42) (3.81) (2.77) (2.43) (1.51) (0.71) R2 1.16 0.48 0.48 0.28 0.31 0.21 0.28 0.26 0.24 0.24

Convergence to Efficiency, April 11, 2004 47 Table 6 (Continued)

Explanatory Return Interval (minutes) Variable Five Ten Fifteen Thirty Sixty Dependent Variable is the Midpoint Returnt, Large Stocks, 2002 -0.050 -0.041 -0.042 -0.040 -0.031 -0.031 -0.011 -0.013 -0.002 -0.005 Midpoint (-6.28) (-5.37) (-3.81) (-3.78) (-2.26) (-2.33) (-0.60) (-0.69) (-0.05) (-0.15) Return t-1 (-3.40) (-2.35) (-2.23) (-2.12) (-1.92) (-1.87) (-0.65) (-0.78) (-0.09) (-0.21) 1.07 0.487 3.74 0.322 0.071 0.061 OIB#t-1 (5.86) (2.05) (2.15) (1.22) (0.37) (0.24) (2.23) (1.65) (2.50) (1.12) (0.27) (0.24) 5.58 2.96 1.20 0.861 OIB$t-1 (4.30) (1.44) (0.56) (0.40) (2.41) (1.93) (0.72) (0.35) R2 0.45 0.34 0.33 0.32 0.24 0.21 0.13 0.13 0.15 0.15 Dependent Variable is the Midpoint Returnt, Mid-cap Stocks, 2002 0.026 -0.007 -0.026 -0.018 -0.022 -0.016 -0.004 -0.003 -0.010 -0.004 Midpoint (-3.30) (-0.91) (-2.33) (-1.70) (-1.64) (-1.27) (-0.20) (-0.16) (-0.36) (-0.15) Return t-1 (-1.88) (-0.50) (1.60) (-1.12) (-1.50) (-1.10) (-0.24) (-0.21) (-0.49) (-0.21) 1.91 0.822 0.680 0.230 0.348 OIB#t-1 (8.25) (2.69) (1.87) (0.48) (0.57) (4.59) (2.65) (2.24) (0.69) (0.91) 84.16 46.70 4.79 2.30 0.030 OIB$t-1 (4.31) (1.66) (1.40) (0.56 (0.15) (4.14) (2.60) (2.48) (0.98) (0.01) R2 0.53 0.25 0.30 0.23 0.25 0.21 0.17 0.14 0.25 0.00 Dependent Variable is the Midpoint Returnt, Small Stocks, 2002 -0.035 -0.016 -0.036 -0.028 -0.021 -0.016 -0.008 -0.006 0.000 0.003 Midpoint (-4.08) (-1.93) (-3.07) (-2.50) (-1.36) (-1.11) (-0.31) (-0.23) (0.02) (0.12) Return t-1 (-2.12) (-1.03) (-2.01) (-1.62) (-1.04) (-0.87) (-0.47) (-0.37) (0.02) (0.18) 1.97 0.322 0.569 0.301 0.273 OIB#t-1 (7.69) (1.22) (1.43) (0.60) (0.49) (5.18) (1.12) (2.27) (1.02) (1.15) 107.25 7.08 5.29 3.24 2.64 OIB$t-1 (4.17) (1.77) (1.12) (0.47) (0.30) (4.75) (2.66) (2.33) (0.92) (0.77) R2 0.64 0.37 0.24 0.35 0.36 0.34 0.25 0.21 0.13 0.15

48 Convergence to Efficiency, April 11, 2004